Optimal. Leaf size=135 \[ -\frac{\csc ^6(c+d x)}{6 a^4 d}+\frac{4 \csc ^5(c+d x)}{5 a^4 d}-\frac{7 \csc ^4(c+d x)}{4 a^4 d}+\frac{8 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{8 \csc (c+d x)}{a^4 d}+\frac{8 \log (\sin (c+d x))}{a^4 d}-\frac{8 \log (\sin (c+d x)+1)}{a^4 d} \]
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Rubi [A] time = 0.0856235, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2707, 88} \[ -\frac{\csc ^6(c+d x)}{6 a^4 d}+\frac{4 \csc ^5(c+d x)}{5 a^4 d}-\frac{7 \csc ^4(c+d x)}{4 a^4 d}+\frac{8 \csc ^3(c+d x)}{3 a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{8 \csc (c+d x)}{a^4 d}+\frac{8 \log (\sin (c+d x))}{a^4 d}-\frac{8 \log (\sin (c+d x)+1)}{a^4 d} \]
Antiderivative was successfully verified.
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Rule 2707
Rule 88
Rubi steps
\begin{align*} \int \frac{\cot ^7(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a-x)^3}{x^7 (a+x)} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^7}-\frac{4 a}{x^6}+\frac{7}{x^5}-\frac{8}{a x^4}+\frac{8}{a^2 x^3}-\frac{8}{a^3 x^2}+\frac{8}{a^4 x}-\frac{8}{a^4 (a+x)}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{8 \csc (c+d x)}{a^4 d}-\frac{4 \csc ^2(c+d x)}{a^4 d}+\frac{8 \csc ^3(c+d x)}{3 a^4 d}-\frac{7 \csc ^4(c+d x)}{4 a^4 d}+\frac{4 \csc ^5(c+d x)}{5 a^4 d}-\frac{\csc ^6(c+d x)}{6 a^4 d}+\frac{8 \log (\sin (c+d x))}{a^4 d}-\frac{8 \log (1+\sin (c+d x))}{a^4 d}\\ \end{align*}
Mathematica [A] time = 0.170585, size = 89, normalized size = 0.66 \[ \frac{-10 \csc ^6(c+d x)+48 \csc ^5(c+d x)-105 \csc ^4(c+d x)+160 \csc ^3(c+d x)-240 \csc ^2(c+d x)+480 \csc (c+d x)+480 \log (\sin (c+d x))-480 \log (\sin (c+d x)+1)}{60 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.148, size = 130, normalized size = 1. \begin{align*} -8\,{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{{a}^{4}d}}-{\frac{1}{6\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{6}}}+{\frac{4}{5\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{5}}}-{\frac{7}{4\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{4}}}+{\frac{8}{3\,{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{3}}}-4\,{\frac{1}{{a}^{4}d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+8\,{\frac{1}{{a}^{4}d\sin \left ( dx+c \right ) }}+8\,{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{{a}^{4}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.68077, size = 128, normalized size = 0.95 \begin{align*} -\frac{\frac{480 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{4}} - \frac{480 \, \log \left (\sin \left (d x + c\right )\right )}{a^{4}} - \frac{480 \, \sin \left (d x + c\right )^{5} - 240 \, \sin \left (d x + c\right )^{4} + 160 \, \sin \left (d x + c\right )^{3} - 105 \, \sin \left (d x + c\right )^{2} + 48 \, \sin \left (d x + c\right ) - 10}{a^{4} \sin \left (d x + c\right )^{6}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59832, size = 502, normalized size = 3.72 \begin{align*} \frac{240 \, \cos \left (d x + c\right )^{4} - 585 \, \cos \left (d x + c\right )^{2} + 480 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac{1}{2} \, \sin \left (d x + c\right )\right ) - 480 \,{\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 16 \,{\left (30 \, \cos \left (d x + c\right )^{4} - 70 \, \cos \left (d x + c\right )^{2} + 43\right )} \sin \left (d x + c\right ) + 355}{60 \,{\left (a^{4} d \cos \left (d x + c\right )^{6} - 3 \, a^{4} d \cos \left (d x + c\right )^{4} + 3 \, a^{4} d \cos \left (d x + c\right )^{2} - a^{4} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.93045, size = 313, normalized size = 2.32 \begin{align*} -\frac{\frac{30720 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{15360 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{4}} + \frac{37632 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 10080 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 2835 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 880 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 240 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 48 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 5}{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6}} + \frac{5 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 48 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 240 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 880 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2835 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 10080 \, a^{20} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{24}}}{1920 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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